A constellation of scientific satellites began sending unexpected data to mission control just before an important observation campaign. Nova and Luna examined the input-output tables for several control commands and compared the function rules governing each satellite's responses. One proposed rule matched only some of the observed outputs, while another correctly predicted every response, including new test commands. After identifying the consistent function, they updated the network settings. The satellites synchronized again, restoring communication and allowing the mission to collect all of its planned scientific observations.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
Substitute several input values into each proposed rule.
One satellite produced the input-output pairs (1, 4), (2, 9), (3, 16), and (4, 25). Which function rule matches every response?
f(x) = x² + 3f(x) = (x + 1)²f(x) = 4xf(x) = x² + 1Correct! The rule f(x) = (x + 1)² produces all four recorded outputs.
Try again. Test each rule with more than one input from the table.
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Hint
Calculate f(3) first, then substitute that result into g.
The network uses the function f(x) = 2x² - 3. A test command first uses input 3, and then the resulting output becomes the input for the function g(x) = x + 4. What is the final output?
Correct! Applying the two functions in order gives a final output of 19.
Apply f first, then use its output as the input of g.
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Hint
Test both rules with all three recorded inputs.
Two proposed satellite rules are f(x) = 3x - 2 and g(x) = x². The recorded responses are (1, 1), (2, 4), and (3, 9). Which conclusion should Nova and Luna make?
Only f matches every recorded response.Only g matches every recorded response.Both f and g match every recorded response.Neither f nor g matches any recorded response.Correct! The rule g(x) = x² matches all three responses, while f does not.
Check every input-output pair rather than testing only one value.
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